254
ANALYTICAL MECHANICS
ILLUSTRATIVE EXAMPLES.
1. A jet of water strikes a concave vessel with a velocity of 80 feet per second and then leaves it with a velocity which has the same magnitude as the velocity of impact but makes an angle of 120° with it. If the diameter of the jet is 1 inch find the force necessary to hold the concave vessel in position.
The force experienced by the vessel equals the rate at which it receives momentum. Suppose the vessel to be symmetrical with respect to the axis of the jet, as in Fig. 119, then by symmetry there can be no resultant force on the vessel in a direction perpendicular to the axis of the jet. Therefore we need to consider only the change in momentum along the axis. Let m be the mass of water delivered by the jet in the time i, v the velocity of impact, and a the change in the direction of flow. Then the force is a X
„ mv — mv cos a -*--
— — v (1 — cos a) t
= — v (1 — cos a)
w\ t* \.
— -1 -- v (1 — cos a)
g t
(1 — cos a),
where A is the area of the cross-section of the jet and w\ is the weight of a cubic foot of water. Replacing the various magnitudes by their numerical values we obtain
')
32
ft. sec.2
= 102.3 Ib.
DISCUSSION. — It is evident from the general expression of F that its value depends upon a. and varies between zero for a = 0 and • Wl v for
a = TT. When a = f, F =